De Finetti theorem on the infinite non-commutative torus

The set of spreadable states on an infinite non-commutative torus A α Z A_α ^Z is determined for all values of the deformation parameter α α . If α 2 π α /2π is irrational, the canonical trace is the only spreadable state. If α 2 π α /2π is rational, the set of all spreadable states is a Bauer simplex. Moreover, its boundary is the set of all infinite products of a single state on C ( T ) C(T) , which is invariant under all rotations by n 0 n_0 -th roots of unity, where n 0 = p 1 { m 1 2 } … p r { m r 2 } n_0=p_1^{\{m_1/2\}} p_r^{\{m_r/2\}} , with { n 2 } \{n/2\} being n 2 n/2 for